Kinetic Molecular Theory

Maxwell Distribution: Concepts

In the context of the Kinetic Molecular Theory, a gas contains a large number of particles in rapid motion. Each particle has a different speed, and each collision between particles changes the speeds of the particles. An understanding of the properties of the gas requires an understanding of the distribution of particle speeds.

The simulation below shows the motion of particles in a gas. Each blue circle represents a neon atom. The histogram at the right of the simulation shows the distribution of speeds. The speed range (along the horizontal axis) of 0 to 5000 m/sec is divided into 50 cells (0 to 100 m/sec, 100 to 200 m/sec, etc.). The height of the bar for a particular cell equals the relative number of neon atoms having a speed in that cell's range.

This histogram depicts the Maxwell Distribution, which is the distribution of particle speeds at a given temperature. The blue curve is the theoretical prediction for the Maxwell Distribution. As data is collected from the simulation over a lengthy period of time, the red bars of the histogram should conform to the shape of the blue curve.

Run the simulation for awhile until the shape of the speed distribution becomes well-defined. After starting the simulation, it takes about 30 to 40 sec for the system to equilibrate before histogram data is plotted.

What is the shape of the distribution? Is it symmetric (as many slow-moving particles as fast-moving particles)?

How fast are the particles moving? For comparison, 70 mph = 31.3 m/sec. The speed of sound near sea level is around 761 mph or 340 m/sec.

Properties of the Maxwell Distribution

The Maxwell distribution describes the distribution of particle speeds in an ideal gas. The distribution may be characterized in a variety of ways.

• Average Speed

  The average speed is the sum of the speeds of all of the particles divided by the number of particles.

• Most Probable Speed

  The most probable speed is the speed associated with the highest point in the Maxwell distribution. Only a small fraction of particles might have this speed, but it is more likely than any other speed.

• Width of the Distribution

  The width of the distribution characterizes the range of speeds for the particles. An easy assessment of the width is the full-width at half-maximum (FWHM). On the left and right sides of the distribution, find the point where the distribtion is one-half of its maximum value. The difference between these two half-maximum positions is the FWHM. For the distribution shown above, the FWHM is 563 m/sec.

The simulation below depicts the behavior of 100 neon atoms at five different temperatures. Run the simulation at each temperature sufficiently long to obtain a reasonably well-defined distribution of speeds.

What happens to the shape of the Maxwell Distribution as the temperature increases?

At each temperature estimate the most probable speed and the width of the distribution.

Carefully observe the behavior of the particles at the various temperatures. How is the particle behavior reflected in the histogram?